Bounds of the Derivative of Some Classes of Rational Functions

Nuttapong Arunrat, Keaitsuda Maneeruk Nakprasit

Abstract

Let $r(z)$ be a rational function with at most $n$ poles, $a_1, a_2, \ldots, a_n,$ where $|a_j| > 1,$ $1\leq j\leq n.$This paper investigates the estimate of the modulus of the derivative of a rational function $r(z)$ on the unit circle. We establish an upper bound when all zeros of $r(z)$ lie in $|z|\geq k\geq 1$ and a lower bound when all zeros of $r(z)$ lie in $|z|\leq  k \leq 1.$ In particular,  when $k=1$ and $r(z)$ has exactly $n$ zeros, we obtain a generalization of results by A. Aziz and W. M. Shah.